VIX Index – Volatility formula explained

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CBOE Volatility Index (VIX) Definition

What Is the CBOE Volatility Index (VIX)?

Created by the Chicago Board Options Exchange (CBOE), the Volatility Index, or VIX, is a real-time market index that represents the market’s expectation of 30-day forward-looking volatility. Derived from the price inputs of the S&P 500 index options, it provides a measure of market risk and investors’ sentiments. It is also known by other names like “Fear Gauge” or “Fear Index.” Investors, research analysts and portfolio managers look to VIX values as a way to measure market risk, fear and stress before they take investment decisions.

Key Takeaways

  • The CBOE Volatility Index, or VIX, is a real-time market index representing the market’s expectations for volatility over the coming 30 days.
  • Investors use the VIX to measure the level of risk, fear, or stress in the market when making investment decisions.
  • Traders can also trade the VIX using a variety of options and exchange-traded products, or use VIX values to price derivatives.

How Does the VIX Work?

For financial instruments like stocks, volatility is a statistical measure of the degree of variation in their trading price observed over a period of time. On 27 September 2020, shares of Texas Instruments Inc. (TXN) and Eli Lilly & Co. (LLY) closed around similar price levels of $107.29 and $106.89 per share, respectively. However, a look at their price movements over the past one month (September) indicates that TXN (Blue Graph) had much wider price swings compared to that of LLY (Orange Graph). TXN had higher volatility compared to LLY over the one-month period.

Extending the observation period to last three months (July to September) reverses the trend: LLY had much wider range for price swings compared to that of TXN, which is completely different from the earlier observation made over one month. LLY had higher volatility than TXN during the three month period.

Volatility attempts to measure such magnitude of price movements that a financial instrument experiences over a certain period of time. The more dramatic the price swings are in that instrument, the higher the level of volatility, and vice versa.

How Volatility is Measured

Volatility can be measured using two different methods. First is based on performing statistical calculations on the historical prices over a specific time period. This process involves computing various statistical numbers, like mean (average), variance and finally the standard deviation on the historical price data sets. The resulting value of standard deviation is a measure of risk or volatility. In spreadsheet programs like MS Excel, it can be directly computed using the STDEVP() function applied on the range of stock prices. However, standard deviation method is based on lots of assumptions and may not be an accurate measure of volatility. Since it is based on past prices, the resulting figure is called “realized volatility” or “historical volatility (HV).” To predict future volatility for the next X months, a commonly followed approach is to calculate it for the past recent X months and expect that the same pattern will follow.

The second method to measure volatility involves inferring its value as implied by option prices. Options are derivative instruments whose price depends upon the probability of a particular stock’s current price moving enough to reach a particular level (called the strike price or exercise price). For example, say IBM stock is currently trading at a price of $151 per share. There is a call option on IBM with a strike price of $160 and has one month to expiry. The price of such a call option will depend upon the market perceived probability of IBM stock price moving from current level of $151 to above the strike price of $160 within the one month remaining to expiry. Since the possibility of such price moves happening within the given time frame are represented by the volatility factor, various option pricing methods (like Black Scholes model) include volatility as an integral input parameter. Since option prices are available in the open market, they can be used to derive the volatility of the underlying security (IBM stock in this case). Such volatility, as implied by or inferred from market prices, is called forward looking “implied volatility (IV).”

Though none of the methods is perfect as both have their own pros and cons as well as varying underlying assumptions, they both give similar results for volatility calculation that lie in a close range.

Extending Volatility to Market Level

In the world of investments, volatility is an indicator of how big (or small) moves a stock price, a sector-specific index, or a market-level index makes, and it represents how much risk is associated with the particular security, sector or market. The above stock-specific example of TXN and LLY can be extended to sector-level or market-level. If the same observation is applied on the price moves of a sector-specific index, say the NASDAQ Bank Index (BANK) which comprises of more than 300 banking and financial services stocks, one can assess the realized volatility of the overall banking sector. Extending it to the price observations of the broader market level index, like the S&P 500 index, will offer a peek into volatility of the larger market. Similar results can be achieved by deducing the implied volatility from the option prices of the corresponding index.

Having a standard quantitative measure for volatility makes it easy to compare the possible price moves and the risk associated with different securities, sectors and markets.

The VIX Index is the first benchmark index introduced by the CBOE to measure the market’s expectation of future volatility. Being a forward looking index, it is constructed using the implied volatilities on S&P 500 index options (SPX) and represents the market’s expectation of 30-day future volatility of the S&P 500 index which is considered the leading indicator of the broad U.S. stock market. Introduced in 1993, the VIX Index is now an established and globally recognized gauge of U.S. equity market volatility. It is calculated in real-time based on the live prices of S&P 500 index. Calculations are performed and values are relayed during 2:15 a.m. CT and 8:15 a.m. CT, and between 8:30 a.m. CT and 3:15 p.m. CT. CBOE began dissemination of the VIX Index outside of U.S. trading hours in April 2020.

Calculation of VIX Index Values

VIX index values are calculated using the CBOE-traded standard SPX options (that expire on the third Friday of each month) and using the weekly SPX options (that expire on all other Fridays). Only those SPX options are considered whose expiry period lies within 23 days and 37 days.

While the formula is mathematically complex, theoretically it works as follows. It estimates the expected volatility of the S&P 500 index by aggregating the weighted prices of multiple SPX puts and calls over a wide range of strike prices. All such qualifying options should have valid non-zero bid and ask prices that represent the market perception of which options’ strike prices will be hit by the underlying during the remaining time to expiry. For detailed calculations with example, one can refer to the section “VIX Index Calculation: Step-by-Step” of the VIX whitepaper.

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Evolution of VIX

During its origin in 1993, VIX was calculated as a weighted measure of the implied volatility of eight S&P 100 at-the-money put and call options, when the derivatives market had limited activity and was in growing stages. As the derivatives markets matured, ten years later in 2003, CBOE teamed up with Goldman Sachs and updated the methodology to calculate VIX differently. It then started using a wider set of options based on the broader S&P 500 index, an expansion which allows for a more accurate view of investors’ expectations on future market volatility. The then adopted methodology continues to remain in effect, and is also used for calculating various other variants of volatility index.

Real World Example of the VIX

Volatility value, investors’ fear and the VIX index values move up when the market is falling. The reverse is true when market advances – the index values, fear and volatility decline.

A real world comparative study of the past records since 1990 reveals several instances when the overall market, represented by S&P 500 index (Orange Graph) spiked leading to the VIX values (Blue Graph) going down around the same time, and vice versa.

One should also note that VIX movement is much more than that observed in the index. For example, when S&P 500 declined around 15% between August 1, 2008 and October 1, 2008, the corresponding rise in VIX was nearly 260%.

In absolute terms, VIX values greater than 30 are generally linked to a large volatility resulting from increased uncertainty, risk and investors’ fear. VIX values below 20 generally correspond to stable, stress-free periods in the markets.

How to Trade the VIX

VIX index has paved the way for using volatility as a tradable asset, although through derivative products. CBOE launched the first VIX-based exchange-traded futures contract in March 2004, which was followed by the launch of VIX options in February 2006. Such VIX-linked instruments allow pure volatility exposure and have created a new asset class altogether. Active traders, large institutional investors and hedge fund managers use the VIX-linked securities for portfolio diversification, as historical data demonstrates a strong negative correlation of volatility to the stock market returns – that is, when stock returns go down, volatility rises and vice versa.

Other than the standard VIX index, CBOE also offers several other variants for measuring broad market volatility. Other similar indexes include the Cboe ShortTerm Volatility Index (VXSTSM) – which reflects 9-day expected volatility of the S&P 500 Index, the Cboe S&P 500 3-Month Volatility Index (VXVSM) and the Cboe S&P 500 6-Month Volatility Index (VXMTSM). Products based on other market indexes include the Nasdaq-100 Volatility Index (VXNSM), Cboe DJIA Volatility Index (VXDSM) and the Cboe Russell 2000 Volatility Index (RVXSM). Options and futures based on RVXSM are available for trading on CBOE and CFE platforms, respectively.

Like all indexes, one cannot buy the VIX directly. Instead investors can take position in VIX through futures or options contracts, or through VIX-based exchange-traded products (ETP). For example, ProShares VIX Short-Term Futures ETF (VIXY), iPath Series B S&P 500 VIX Short Term Futures ETN (VXXB) and VelocityShares Daily Long VIX Short-Term ETN (VIIX) are many such offerings which track certain VIX-variant index and take positions in linked futures contracts.

Active traders who employ their own trading strategies as well as advanced algorithms use VIX values to price the derivatives which are based on high beta stocks. Beta represents how much a particular stock price can move with respect to the move in broader market index. For instance, a stock having a beta of +1.5 indicates that it is theoretically 50% more volatile than the market. Traders making bets through options of such high beta stocks utilize the VIX volatility values in appropriate proportion to correctly price their option trades.

Tracking Volatility

When market volatility spikes or stalls, financial websites, bloggers, social media, newspapers and television commentators all refer to the VIX ® . Formally known as the CBOE Volatility Index, the VIX is a benchmark index designed specifically to track S&P 500 volatility. Most investors familiar with the VIX commonly refer to it as the “fear gauge,” because it has become a proxy for market volatility.

Key Takeaways

  • The VIX is a benchmark index designed specifically to track S&P 500 volatility.
  • The VIX is calculated using a formula to derive expected volatility by averaging the weighted prices of out-of-the-money puts and calls.
  • Volatility is useful to investors, as it gives them a way to gauge the market environment; it also provides investment opportunities.

The VIX was created by the Chicago Board Options Exchange (CBOE), which bills itself as “the largest U.S. options exchange and creator of listed options.” The CBOE runs a for-profit business selling (among other things) investments to sophisticated investors. These include hedge funds, professional money managers and individuals that make investments seeking to profit from market volatility. To facilitate and encourage these investments, the CBOE developed the VIX, which tracks market volatility on a real-time basis.

While the math behind the calculation and the accompanying explanation takes up most of a 15-page white paper published by the CBOE, we’ll provide the highlights in an overview. Here’s a look at the calculations behind the VIX, courtesy of examples and information provided by the CBOE.


A Look at the VIX for the Mildly Curious

The CBOE provides the following formula as a general example of how the VIX is calculated:

The calculations behind each part of the equation are rather complex for most people who don’t do math for a living. They are also far too complex to fully explain in a short article, so let’s put some numbers into the formula to make the math easier to follow:

Delving into the Details of the Volatility Index

The VIX is calculated using a “formula to derive expected volatility by averaging the weighted prices of out-of-the-money puts and calls.” Using options that expire in 16 and 44 days, respectively, in the example below, and starting on the far left of the formula, the symbol on the left of “=” represents the number that results from the calculation of the square root of the sum of all the numbers that sit to the right multiplied by 100.

To get to that number:

  1. The first set of numbers to the right of the “=” represents time. This figure is determined by using the time to expiration in minutes of the nearest term option divided by 525,600, which represents the number of minutes in a 365-day year. Assuming the VIX calculation time is 8:30 a.m., the time to expiration in minutes for the 16-day option will be the number of minutes within 8:30 a.m. today and 8:30 a.m. on the settlement day. In other words, the time to expiration excludes midnight to 8:30 a.m. today and excludes 8:30 a.m. to midnight on the settlement day (full 24 hours excluded). The number of days we’ll be working with will technically be 15 (16 days minus 24 hours), so it’s 15 days x 24 hours x 60 minutes = 21,600. Use the same method to get the time to expiration in minutes for the 44-day option to get 43 days x 24 hours x 60 minutes = 61,920 (Step 4).
  2. The result is multiplied by the volatility of the option, represented in the example by 0.066472.
  3. The result is then multiplied by the result of the difference between the number of minutes to expiration of the next term option (61,920) minus the number of minutes in 30 days (43,200). This result is divided by the difference of the number of minutes to expiration of the next term option (61,920) minus the number of minutes to expiration of the near term option (21,600). Just in case you’re wondering where 30 days came from, the VIX uses a weighted average of options with a constant maturity of 30 days to expiration.
  4. The result is added to the sum of the time calculation for the second option, which is 61,920 divided by the number of minutes in a 365-day year (526,600). Just as in the first calculation, the result is multiplied by the volatility of the option, represented in the example by 0.063667.
  5. Next we repeat the process covered in step 3, multiplying the result of step 4 by the difference of the number of minutes in 30 days (43,200), minus the number of minutes to expiration of the near-term options (21,600). We divide this result by the difference of the number of minutes to expiration of the next-term option (61,920) minus the number of minutes to expiration of the near-term options (21,600).
  6. The sum of all previous calculations is then multiplied by the result of the number of minutes in a 365-day year (526,600) divided by the number of minutes in 30 days (43,200).
  7. The square root of that number multiplied by 100 equals the VIX.

Clearly, the order of operations is critical in the calculation and, for most of us, calculating the VIX isn’t the way we would choose to spend a Saturday afternoon. And if we did, the exercise would certainly take up most of the day. Fortunately, you will never have to calculate the VIX because the CBOE does it for you. Thanks to the Internet, you can go online, type in the ticker VIX and get the number delivered to your screen in an instant.

Investing in Volatility

Volatility is useful to investors, as it gives them a way to gauge the market environment. It also provides investment opportunities. Since volatility is often associated with negative stock market performance, volatility investments can be used to hedge risk. Of course, volatility can also mark rapidly rising markets. Whether the direction is up or down, volatility investments can also be used to speculate.

As one might expect, investment vehicles used for this purpose can be rather complex. VIX options and futures provide popular vehicles through which sophisticated traders can place their hedges or implement their hunches. Professional investors use these on a routine basis.

Exchange-traded notes – a type of unsecured, unsubordinated debt security – can also be used. ETNs that track volatility include the iPath S&P 500 VIX Short-Term Futures (VXX) and the VelocityShares Daily Inverse VIX Short-Term (XIV).

Exchange-traded funds offer a somewhat more familiar vehicle for many investors. Volatility ETF options include the ProShares Ultra VIX Short-Term Futures (UVXY) and ProShares VIX Mid-Term Futures (VIXM).

There are pros and cons to each of these investment vehicles that should be thoroughly evaluated before making investment decisions.

The Bottom Line

Regardless of purpose (hedging or speculation) or the specific investment vehicles chosen, investing in volatility is not something to jump into without taking some time to understand the market, the investment vehicles and the range of possible outcomes. Failing to do the proper preparation and taking a prudent approach to investing can have a more detrimental result on your personal bottom line than making a mathematical error in your VIX calculation.

VIX Calculation Explained

The objective of this page is to explain the logic of VIX calculation and some of the underlying assumptions and parameters. Exact formulas are available in a short pdf named VIX White Paper on the official website of CBOE.

If you are not familiar with VIX, you may first want to see a more basic explanation: What is VIX?

VIX Calculation: The Big Picture

VIX is interpreted as annualized implied volatility of a hypothetical option on S&P500 with 30 days to expiration, based on the prices of near-term S&P500 options traded on CBOE.

Contrary to what many people believe, the VIX is not calculated using Black-Scholes or any other option pricing model. There is a formula which directly derives variance from the whole set of prices of options with the same time to expiration. Two different variances for two different expirations are then interpolated to get 30-day variance. This variance is then transformed into standard deviation (by taking the square root) and multiplied by 100.

VIX Calculation Step by Step

  • Select the options to be included in VIX calculation – a range of call and put strikes in two consecutive expirations around the target 30-day mark.
  • Calculate each option’s contribution to the total variance of its expiration.
  • Calculate the total variance for the first and the second expiration.
  • Calculate 30-day variance by interpolating the two variances, depending on the time to expiration of each.
  • Take the square root to get volatility as standard deviation.
  • Multiply the volatility (standard deviation) by 100.
  • The result is VIX.

The rest of this page explains individual steps in more detail.

Options Included in VIX Calculation

Expirations included

The data used for VIX calculation are bid and ask quotes of short term S&P500 options. Because the target time horizon for the VIX index is 30 days, two consecutive expirations with more than 23 days and less than 37 days are used. These can include the standard monthly expirations as well as weekly S&P 500 options.

The two expirations are referred to as “near-term” and “next-term”. As soon as the near-term options get less than 24 days to expiration, they are no longer used. The previously next-term expiration becomes the new near-term expiration and the next available expiration is added as the new next-term. This rollover happens every week.

Strike prices included

At the money and out of the money call and put options enter VIX calculation and only options which have non-zero bid are included. This is to eliminate illiquid far out of the money options which can imply extreme values of volatility and therefore distort the final VIX value. The selection of strikes goes from the at the money strike up (for calls) and down (for puts), until two consecutive strikes with zero bid price are found in each direction. No other options beyond such two consecutive zero bid strikes are included.

As a result, the range and the total number of options included in VIX calculation vary over time, in line with changes in S&P500 index value and changes in quotes on individual S&P500 options.

Only S&P500 option quotes directly from CBOE are used.

Parameters Used in VIX Calculation

Expected variance of each expiration month is derived from a set of option prices and strikes, given time to expiration and risk-free interest rate.

Time to expiration

The time to expiration for a particular option is calculated very precisely in minutes. The end of the period is the moment when the exercise-settlement value is being determined, which is the open (8:30 am Chicago time) on the settlement day for monthly S&P500 options (usually the third Friday of a month) and close of trading (3:00 pm) for weekly options.

Risk-free interest rate

The interest rate used in VIX calculation is the bond-equivalent yield of US T-bills which mature closest to the particular option expiration. Different interest rates may be used for the two different expirations which enter VIX calculation.

Contributions of Individual Options

The contribution of individual options to the calculation of total variance of an expiration depends on the option’s price, the strike price, and the average strike price increment of neighboring strikes. In general, at the money options influence the final result the most and the contributions decrease as you go further out of the money.

Getting the 30-day Variance from the Two Months

The 30-day variance is calculated by interpolating the total variances of the two expirations. The weights of the two variances depend on how close or far each expiration is from the desired 30-day mark (the closer, the greater weight). The sum of the weights was always 1.

Until October 2020 when only monthly expirations were used, if both expiration months had more than 30 days left (e.g. 32 and 67 days), the first month’s weight was greater than 1 and the second month’s weight was negative.

Calculating the VIX: Final Steps

Having calculated the 30-day variance, we then need to take the square root to transform variance into standard deviation (which is the traditional way how volatility is quoted and VIX is no exception).

The last step is to multiply the result by 100. While volatility usually is in percent, the VIX is volatility times 100. For example, if VIX is 22, it means that a hypothetical S&P500 option with 30 days to expiration has annualized implied volatility of 22%.

Old VIX Calculation Methods

22 September 2003 – 5 October 2020

Until October 2020, the VIX calculation used monthly options only. The rule was two nearest monthly expirations with at least one week left to expiration. For example, if the nearest expirations were in 4, 32, and 67 days, the front month (4 days to expiration) wouldn’t be included, and the next two months (32 and 67 days) would be used in VIX calculation. This was to eliminate options in the last days before expiration, whose prices sometimes behave in strange ways.

Once weekly S&P 500 options became liquid enough, it was logical for CBOE to start using them from 6 October 2020. This made the window around the 30 days target narrower and the calculation more precise.

The old version of the VIX using monthly options only is still being calculated and available under the symbol VIXMO.

Before 22 September 2003

Until September 2003 the VIX was calculated in an entirely different way, even using a different underlying:

  • Until September 2003 S&P100 (OEX) option prices were used. The current method uses S&P500 (SPX) options.
  • The old method used at the money options only. Under the new method, a wide range of strikes enters VIX calculation.
  • The exact way how volatility is derived from option prices is different. The old method used an option pricing model. The new method uses a direct formula.

As you can see, the change in 2003 was much more significant then the one in 2020. The pre-2003 method index is still being calculated and published by CBOE under the ticker symbol VXO. The two methods of course produce different index values, although the differences are not that big and the two indices (VIX and VXO) react to the same market conditions in a similar way.

Daily historical data is available starting from 1990 for the VIX, from 1986 for VXO (therefore VXO data covers the very interesting events of October 1987).

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